FEM_BASIS_T6_DISPLAY
Display a T6 Finite Element Basis Function

FEM_BASIS_T6_DISPLAY is a MATLAB program, using double precision arithmetic, which reads data defining a quadratic triangular ("T6") mesh and displays the basis function associated with any node in the mesh.

The entire mesh is shown (which could be a problem for large meshes!). However, a finite element basis function is nonzero over only a relatively small number of elements. Over these elements, the basis function is quadratic, and the curved nature of this part of the surface is suggested by taking many plot points in this region.

The best part of the presentation is that MATLAB allows the user to turn the plot or zoom in and out, which makes it much easier to understand the information that is being presented.

The triangular mesh is defined by a node file containing the coordinates of nodes, and a triangle file containing sets of the six nodes that make up each triangle.

Note that the six nodes making up each triangle are to be listed in a particular order, in which the vertices come first, followed by the midside nodes. Moreover, the first midside node is the one between vertices 1 and 2, as suggested by the following diagram:

        3
        |\
        | \
        |  \
        6   5
        |    \
        |     \
        |      \
        1---4---2
      

In general, a mesh of quadratic triangles allows the sides to be bent. However, for this application, it is assumed that the sides of each triangle are straight, that is, that each midside node lies on the line segment joining the corresponding vertices.

Usage:

fem_basis_t6_display ( node_file, element_file )

reads the triangular mesh defined by node_file and element_file, queries the user for the node index that defines the basis function, and then displays the result.

Related Programs:

BEZIER_SURFACE_DISPLAY is a MATLAB program which displays a Bezier surface.

FEM_50 is a MATLAB finite element program in just 50 lines of code.

FEM_50_HEAT is a modified version of FEM_50 suitable for solving the heat equation.

FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a linear triangle ("T3") mesh.

FEM_IO is a library of MATLAB routines for reading or writing the node, element and data files that define a finite element model.

FEM_SAMPLE is a MATLAB library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

FEM2D_HEAT is a MATLAB program that solves the time dependent heat equation in the unit square.

FEM2D_POISSON is a MATLAB program for solving Poisson's equation on a square, using the finite element method.

FEMPACK is a library of MATLAB routines for simple finite element calculations.

FREE_FEM_HEAT is a MATLAB program that solves the time dependent heat equation in an arbitrary triangulated 2D region.

FREE_FEM_NAVIER_STOKES is a MATLAB program for solving the steady incompressible Navier Stokes equations on an arbitrary triangulated region, using the finite element method.

FREE_FEM_POISSON is a MATLAB program for solving Poisson's equation on a triangulated region, using the finite element method.

FREE_FEM_STOKES is a MATLAB program for solving the Stokes equations on a triangulated region, using the finite element method.

POLYGONAL_SURFACE_DISPLAY is a MATLAB program which reads two files defining a polygonal surface and displays it within MATLAB.

TET_MESH_DISPLAY is a MATLAB program which can read in the node and tetra files defining a tet mesh and display a wireframe image.

TRI_MESH_DISPLAY is a MATLAB program which reads data defining a triangular mesh of a 3D surface and displays it.

TRIANGLE is an executable C program which computes a triangulation of a geometric region.

TRIANGULATION_DISPLAY_OPEN_GL is an executable C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_ORDER6 is a directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_ORDER6_CONTOUR is a MATLAB script which can make contours of a scalar quantity defined on the nodes of an order 6 triangulation.

TRIANGULATION_PLOT is a MATLAB program which creates an EPS image of a triangulation.

Reference:

1. Klaus-Juergen Bathe,
   Finite Element Procedures,
   Prentice Hall, 1996.
2. Eric Becker, Graham Carey, John Oden,
   Finite Elements, An Introduction, Volume I,
   Prentice-Hall, 1981.
3. Herbert Edelsbrunner,
   Geometry and Topology for Mesh Generation,
   Cambridge, 2001,
   LC: QA377.E36,
   ISBN: 0-521-79309-2.
4. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989,
   LC: TA357.G86.
5. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Prentice Hall, 1973,
   LC: TA335.S77.

Source code:

• basis_11_t6.m evaluates one of the basis functions associated with a quadratic triangular element.
• fem_basis_t6_display.m gets information about a mesh, and the node for which the corresponding basis function is desired, and displays a plot.
• i4_modp.m performs a modulus function, always with a positive result.
• i4_wrap.m uses "wrapping" to force a value to lie within a given range.
• s_escape_tex.m de-escapes TeX escape strings.
• timestamp.m prints the current YMDHMS date as a time stamp.

Tests and Examples

SMALL_SQUARE is a simple mesh of 25 nodes and 8 elements over the [0,4] by [0,4] square. Test files you may copy include:

• small_square_nodes.txt, the node file.
• small_square_elements.txt, the element file.
• small_square_mesh.png, a PNG image of the mesh.
• small_square_node02.png, a PNG image of the basis function associated with node 2.
• small_square_node03.png, a PNG image of the basis function associated with node 3.
• small_square_node13.png, a PNG image of the basis function associated with node 13.

BIG_SQUARE is a simple mesh of 169 nodes and 72 elements over the [0,1] by [0,1] square. Test files you may copy include:

• big_square_nodes.txt, the node file.
• big_square_elements.txt, the element file.
• big_square_mesh.png, a PNG image of the mesh.
• big_square_node84.png, a PNG image of the basis function associated with node 84.
• big_square_node85.png, a PNG image of the basis function associated with node 85.

You can go up one level to the MATLAB source codes.